Optimal. Leaf size=56 \[ \frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {B \log (a+b \cosh (x))}{b} \]
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Rubi [A] time = 0.13, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4401, 2659, 208, 2668, 31} \[ \frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {B \log (a+b \cosh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 208
Rule 2659
Rule 2668
Rule 4401
Rubi steps
\begin {align*} \int \frac {A+B \sinh (x)}{a+b \cosh (x)} \, dx &=\int \left (\frac {A}{a+b \cosh (x)}+\frac {B \sinh (x)}{a+b \cosh (x)}\right ) \, dx\\ &=A \int \frac {1}{a+b \cosh (x)} \, dx+B \int \frac {\sinh (x)}{a+b \cosh (x)} \, dx\\ &=(2 A) \operatorname {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )+\frac {B \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cosh (x)\right )}{b}\\ &=\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {B \log (a+b \cosh (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 55, normalized size = 0.98 \[ \frac {B \log (a+b \cosh (x))}{b}-\frac {2 A \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 291, normalized size = 5.20 \[ \left [\frac {\sqrt {a^{2} - b^{2}} A b \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) + b}\right ) - {\left (B a^{2} - B b^{2}\right )} x + {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{2} b - b^{3}}, -\frac {2 \, \sqrt {-a^{2} + b^{2}} A b \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (B a^{2} - B b^{2}\right )} x - {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{2} b - b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 60, normalized size = 1.07 \[ \frac {2 \, A \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}}} - \frac {B x}{b} + \frac {B \log \left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 137, normalized size = 2.45 \[ -\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}-\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -a -b \right ) a B}{b \left (a -b \right )}-\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -a -b \right ) B}{a -b}+\frac {2 A \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.98, size = 197, normalized size = 3.52 \[ \frac {2\,\mathrm {atan}\left (\frac {A^2\,b^2\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}}{\left (A\,b^3-A\,a^2\,b\right )\,\sqrt {A^2}}+\frac {A^2\,a\,b\,\sqrt {b^2-a^2}}{\left (A\,b^3-A\,a^2\,b\right )\,\sqrt {A^2}}\right )\,\sqrt {A^2}}{\sqrt {b^2-a^2}}-\frac {B\,x}{b}+\frac {B\,b^3\,\ln \left (4\,A^2\,b+8\,A^2\,a\,{\mathrm {e}}^x+4\,A^2\,b\,{\mathrm {e}}^{2\,x}\right )}{b^4-a^2\,b^2}-\frac {B\,a^2\,b\,\ln \left (4\,A^2\,b+8\,A^2\,a\,{\mathrm {e}}^x+4\,A^2\,b\,{\mathrm {e}}^{2\,x}\right )}{b^4-a^2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.72, size = 741, normalized size = 13.23 \[ \begin {cases} \tilde {\infty } \left (2 A \operatorname {atan}{\left (\tanh {\left (\frac {x}{2} \right )} \right )} + B x - 2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} + B \log {\left (\tanh ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\- \frac {A}{b \tanh {\left (\frac {x}{2} \right )}} + \frac {B x}{b} - \frac {2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{b} + \frac {2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} \right )}}{b} & \text {for}\: a = - b \\\frac {A x + B \cosh {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {A \tanh {\left (\frac {x}{2} \right )}}{b} + \frac {B x}{b} - \frac {2 B \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{b} & \text {for}\: a = b \\- \frac {A b \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {A b \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {B a x \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {B a \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {B a \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {2 B a \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {B b x \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {B b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (- \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} - \frac {B b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\sqrt {\frac {a}{a - b} + \frac {b}{a - b}} + \tanh {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} + \frac {2 B b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{a b \sqrt {\frac {a}{a - b} + \frac {b}{a - b}} - b^{2} \sqrt {\frac {a}{a - b} + \frac {b}{a - b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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